Three-Dimensional Solvable Artin Representations Ramified at One Prime

TL;DR

This paper classifies three-dimensional solvable Artin representations ramified at a single prime, using group classification to bound their number based on the Artin conductor.

Contribution

It provides a classification and bounds for three-dimensional solvable Artin representations ramified at one prime, extending understanding of their structure and quantity.

Findings

Classified fixed fields of kernels for these representations

Bounded the number of such representations with a given Artin conductor

Used finite subgroup classification of PGL_3(C)

Abstract

We classify the possibilities for the fixed field of the kernel of an irreducible three-dimensional Artin representation of $\Q$ with solvable image ramified at one prime by using the classification of the finite irreducible subgroups of $\PGL_3(\C)$. This allows us to bound the number of such representations with given Artin conductor.